Elliptic equations, principal eigenvalue and dependence on the domain
José M. Arrieta · Communications in Partial Differential Equations · 1996
We consider a general second order uniformly elliptic differential operator L and also the set θ of all open sets (not necessarily smooth) in the unit ball of ℝn. We define a metric d in this space (up to an equivalence relation ∼) that makes the space (θ/ ∼,d) a complete metric space. We show that the principal eigenvalue and eigenfunction of L are continuous with the metric d. Similar results are obtained for the solutions of the equation Lv = ƒ.